From pictures, the print and your description is the best I can do without being there. I would change out the thermostat, it just might fix your problem.
It could be the thermostat and do you know how to use a voltmeter? At the control board in the compressor unit, at the top of the board are the thermostat wires. Y1 (usually a yellow wire) is the cool wire. The thermostat brings 24vac to this terminal. When the unit won't start after a about a three-five minute wait, check if there is voltage on Y1. Most digital thermostats have a built in timer between compressor cycles. If no voltage is at Y1 on a restart, the problem is the thermostat. If there is voltage and no start then it looks more like the control board but since you power cycle the the air handler, the thermostat gets its 24vac from.
However, I do not see a transformer in the picture. The wire nuts in the small box below are low voltage thermostat wires and the board maybe powered by the air handler transformer, therefore, when you power cycle it, the compressor board is also powered cycled and the problem may clear.
Another test that the compressor board is good, if you look at the wiring, a wire from the 24v transformer runs to the thermostat terminals, that red wire will have 24v ac on it. With a jumper wire short the red to the Y1 and the compressor should kick on. Basic compressor units with a standard control board has all its 24vac from the transformer in the air handler. The red wire terminal usually will have the red wire from the thermostat.
The last two pages is the wiring diagram. As usually it is hard wire print, not much about the board itself. You're right but the contactor is on the board. I think it is the blue relay but and with standard contactor you can easily push the contacts closed to manually start (test) the unit. A 24 volt transformer is also in the compressor unit, at least from the wiring diagram. I believe it is only for the circuit board and the thermostat the 24vac from air handler transformer.